Localised and delocalised states

The Coulomb well potential of an individual atom has sharp energy levels - the states with
different main quantum number n. At the top of the well, where the potential becomes
very shallow, the distinct energy levels approach the continuum characteristic of a free electron.

When many such Coulomb wells are placed in a regular array (i.e. a crystalline lattice), two
effects are observed. First, the potential barrier between adjacent atoms is lowered, allowing the
electrons in states higher up the energy scale to delocalise across the crystal. These electrons
have not enough energy to reach one of the continuum states outside the crystal, but their energy
is sufficient to overcome the internal potential barrier between the atoms. This is exactly the same
observation as in the case of forming a molecule from two Coulomb wells.
Second, the sharp energy levels of the individual Coulomb well are replaced by energy
bands. These bands are getting wider as the energy increases.

Energy bands

We have seen that, when linear-combining states, the number of states coming out of the process is always
equal to the number of states put in. When forming a chemical bond between two atoms, two molecular
states are generated (one bonding and one anti-bonding). By extension it follows that when four atomic
states are combined, four molecular states ensue. These spread over the same range of energies; therefore
the energy splitting between the states is smaller. Again, half the molecular states are bonding and
the other half anti-bonding.

With many atoms in a crystal all contributing to the "molecular" states of the solid, there will be as
many states as atoms contributing. All these states are squeezed into the same energy range as two or
four states were previously. The density of states is so high that it is impossible to distinguish
them: we have an energy band instead. The bonding states are at the bottom of the band, the anti-bonding
ones at the top.

There is also another argument for the existence of energy bands. It is based on the uncertainty principle
between energy and time, .
The tunnelling probability between adjacent Coulomb wells is much lower for the states at the bottom
of the wells (i.e. localised states) as the potential barrier is higher for electrons occupying
those states than for those higher up. Taking the time an electron stays localised in the same Coulomb
well as a measure for the time uncertainty, we can infer that the energy uncertainty for these localised
electrons is rather smaller than for the delocalised ones at the top of the Coulomb potential. This
qualitative argument confirms that the width of the energy bands increases with the energy itself.

Metals and insulators

Consider a material where all states with a certain n are fully occupied with two electrons and all
states at higher n are empty. When adding many atoms together, this results in a completely
filled band at the lower n and an entirely empty one at the higher n. The cutoff between
occupied and unoccupied states (the Fermi level, marked by an orange horizontal line in the Fig.)
is between the two bands, i.e. within the bandgap. The upper band (the
conduction band) is empty, the lower band (the valence band) is completely filled.
Electrons can't move because there is no space for them to do so. This material is an
insulator.

In another material, where the higher-n state is occupied by one electron, the corresponding
band structure has a partially filled band, i.e. the Fermi level is within a band. In this case,
the electrons can easily move as there are plenty of states available within the band at (almost) the same
energy. This material is a metal.

A semiconductor is an insulator with a rather narrow bandgap, enabling some electrons to cross
from the valence band into the conduction band after thermal activation. In an insulator, the temperature
needed to excite significant numbers of electrons into the conduction band would melt the crystal; in a
semiconductor, thermal activation is noticeable at room temperature or even below. Both electrons in the
conduction band and the holes they leave behind in the valence band when making the transition
are mobile and contribute to the conductivity. Sometimes both electrons and holes are coupled by their
Coulomb attraction; they can then be treated as one complex particle, an exciton. More on this
in 3rd year Solid State Physics.

We have seen that each state can be occupied by up to two electrons. To distinguish them, we need another
quantum number which does not follow immediately from the solution of the hydrogen atom:
spin.

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